determining the roots of a polynomial equation,if one divider is given

Question

We are given a polynomial equation $P(x)$, and we know that if we divide the given equation with polynomial $(x^2-1)$ the remainder is $0$. The question is, what can we tell about the roots of $P(x)$? Thank you so much, any advice would be helpful.

Answer 1

This tell you only two solution of $P(x)$ that you can easily understand by factoring $x^2-1$ in $(x+1)(x-1)$: $x=1$ or $x=-1$. For example consider the polynomial $P(x)=(x^2-1)(x-2)(x+9)$: the remainder, dividing by $x^2-1$ is 0, but the solutions are: $x=1$ or $x=-1$ or $x=2$ or $x=-9$.

Answer 2

The exact division means $P(x)=(x^2-1)Q(x)$ where $Q(x)$ is another polynomial. Since $x^2-1=(x+1)(x-1)$, we know that $P(x)$ has $+1$ and $-1$ as two of its roots, but we cannot infer anything else.